How can I manually fix nonplanar geometry?
I will compile multiple methods here from various sources.

The methods for a single face will work for multiple faces if you choose 
Individual Origins as the Pivot center for rotating/scaling. However, this will only work for faces which aren't adjacent. If they share common edges, the single face options will treat them as one large face.

Scale along normal (single face)

Select the vertices of a face, choose the Normal Transform Orientation, then scale along the Normal Z axis. You can do this with the scale manipulator or by pressing SZZ. You can make a single face planar by scaling to 0 with SZZ0.


LoopTools Flatten (single face)

If not already, activate the preinstalled LoopTools Addon. Select the face and choose the Fatten operation from the LoopTools addon. It is available in the 3DView toolshelf T under tools > LoopTools or in the specials menu W > LoopTools Flatten.


Make Planar Faces (multiple faces)

From this answer, there is a tool to make faces planar.


Shift single Vertex

In this answer is a script to make a face planar by only changing a single vertex. Select the three vertices of a face, which shouldn't change, then Shift select the vertex which should be shifted, marking it as the active vertex.
Then, call the Flatten Vertex operator (see the linked answer for details).

Note, that this will not update the Mesh Analysis display. But that will update after a Zero operation, like G0.